This is a preprint of an article accepted for publication in Journal of Engineering Structures on 12 April 2010.

The published article is available online at http://www.sciencedirect.com/science/article/pii/S0141029610001483.

To be cited as: Miquel B., Bouaanani N. 2010. Simplified evaluation of the vibration period and seismic response
of gravity dam-water systems. Journal of Engineering Structures, 32: 2488-2502.

Simplified evaluation of the vibration period

and seismic response of gravity dam-water systems
Benjamin Miquel 1 and Najib Bouaanani 2

ABSTRACT

This paper proposes a practical procedure for a simplified evaluation of the fundamental vibration period of dam-
water systems, and corresponding added damping, force and mass, all key parameters to assess the seismic behavior.
The proposed technique includes the effects of dam geometry and flexibility, dam-reservoir interaction, water com-
pressibility and varying reservoir level. The mathematical derivations of the method are provided considering both
incompressible and compressible water assumptions. In the former case, we propose a closed-form expression for
the fundamental vibration period of a dam-reservoir system. When water compressibility is included, we show that
the fundamental vibration period can be obtained by simply solving a cubic equation. The proposed procedure is
validated against classical Westergaard added mass formulation as well as other more advanced analytical and fi-
nite element techniques. Gravity dam monoliths with various geometries and rigidities impounding reservoirs with
different heights are investigated. The new approach yields results in excellent agreement with those obtained when
the reservoir is modeled analytically, or numerically using potential-based finite elements. The analytical expres-
sions developed and the procedure steps are presented in a manner so that calculations can be easily implemented
in a spreadsheet or program for simplified and practical seismic analysis of gravity dams.

KEY WORDS: Gravity dams; Dam-reservoir systems; Fluid-structure interaction; Analytical formulations;
Finite elements; Dam safety; Vibration period; Earthquake response; Simplified methods.

1 Graduate Research Assistant, Department of Civil, Geological and Mining Engineering,

École Polytechnique de Montréal, Montréal, QC H3C 3A7, Canada.
2 Associate Professor, Department of Civil, Geological and Mining Engineering,

École Polytechnique de Montréal, Montréal, QC H3C 3A7, Canada

Corresponding author. E-mail: najib.bouaanani@polymtl.ca
Nomenclature

Abbreviations

ESDOF Equivalent single degree of freedom

FRF Frequency response function

Roman symbols

A1 , A2 , A3 , A4 coefficients given by Eqs. (59) to (63)

a1 , a2 , a3 coefficients used for cubic approximation of structural mode shapes

B0 , B1 hydrodynamic parameters given by Eqs. (22) and (23), respectively

B0n , B1n hydrodynamic parameters given by Eqs. (24) and (25), respectively

Bb0n , Bb1n hydrodynamic parameters given by Eqs. (32) and (33), respectively

Cn , Cen nth generalized damping of the dam and dam-reservoir system, respectively

Cr velocity of pressure waves in the reservoir

D 1 , D2 coefficients given by Eq. (65)

Es modulus of elasticity of the dam

Fst total hydrostatic force exerted on dam upstream face

Fn , Gn functions given by Eq. (34)

f1 equivalent lateral force given by Eq. (80)

fsc equivalent lateral force including higher mode effects as given by Eq. (83)

Hr , Hs reservoir and dam heights, respectively

Ijn integral given by Eq. (8)

K1 generalized stiffness of the dam at fundamental vibration mode

e
Ln , L nth generalized forces of the dam and dam-reservoir system, respectively
n

M mass matrix of the dam monolith

Ms total mass of the dam monolith

2
mi Westergaard added mass at node i of the dam finite element mesh

f
Mn , M nth generalized masses of the dam and dam-reservoir system, respectively
n

Nr , Ns number of considered reservoir and structural modes, respectively

Q̄, Q̄n vector in Eq. (11) and its elements given by Eq. (13), respectively

p, p̄ hydrodynamic pressure and corresponding FRF, respectively

p̄0 , p̄j hydrodynamic pressure FRFs given by Eq. (3)

p̄0n , p̄jn hydrodynamic pressure FRFs given by Eqs. (4) and (5), respectively

pb̄0 real-valued hydrodynamic pressure given by Eq. (84)

R1 , Rr frequency ratios given by ω1 /ω0 and ωr /ω0 , respectively

S̄, S̄nj matrix in Eq. (11) and its elements given by Eq. (12), respectively

Sa pseudo-acceleration ordinate of the earthquake design spectrum

t time

T1 , Tr fundamental periods of the dam and dam-reservoir system, respectively

U coefficient given by Eq. (67)

ū, ü¯ FRFs for horizontal displacement and acceleration, respectively

V coefficient given by Eq. (67)

Vi volume of water tributary to node i of the dam finite element mesh

v̄, v̈¯ FRFs for vertical displacement and acceleration, respectively

ẍg , ẍ(max)
g ground acceleration time history and peak ground acceleration, respectively

yi height of node i of the dam finite element mesh

Z̄, Z̄j vector of generalized coordinates and j th generalized coordinate, respectively

Greek symbols

γi , γbi coefficients given in Table 1 for i = 1 . . . 6

Γ variable given by Eq. (65)

3
Γ1 , Γ2 , Γ3 , Γ4 analytical solutions of Eq. (64) as given by Eq. (66)

Γ∗ real solution of Eq. (64)

∆ discriminant of Eq. (64)

δnj Kronecker symbol

ε error estimator

ζi , ζbi coefficients given in Table 2 for i = 1 . . . 3

η ratio of reservoir level to dam height, i.e. Hr /Hs

b Θ
θ, θ, parameters given by Eqs. (76), (43) and (42), respectively

κn function given by Eq. (7)

λn nth reservoir eigenvalue

µs mass of the dam per unit height

ν Poisson’s ratio of dam concrete

ξn nth fraction of critical damping of the dam

ξer equivalent damping ratio of the dam-reservoir ESDOF system

ρr , ρs mass densities of water and dam concrete, respectively

τ coefficient given by Eq. (67)

ϕ, ϕ,
b Φ parameters given by Eqs. (57), (39) and (38), respectively

χ frequency parameter defined by Rr2

(x)
ψ n , ψj nth structural mode shape and x–component of the j th structural mode shape

ω exciting frequency

ω0 fundamental vibration frequency of the full reservoir

ωn nth vibration frequency of the dam

ωr fundamental vibration frequency of the dam-reservoir system

4
1 Introduction

Considering the effects of fluid-structure dynamic interactions is important for the design and safety
evaluation of earthquake-excited gravity dams. Significant research has been devoted to this subject
since the pioneering work of Westergaard [1] who modeled hydrodynamic loads as an added-mass at-
tached to the dam upstream face. Although Wesregaard’s analytical formulation was developed assum-
ing a rigid dam impounding incompressible water, it has been widely used for many decades to design
earthquake-resistant concrete dams because of its simplicity. During the last four decades, several re-
searchers developed advanced analytical and numerical approaches to account for dam deformability
and water compressibility in the seismic response of concrete dams [2–12]. Most of these methods are
based on a coupled field solution through sub-structuring of the dam-reservoir system, making use of
analytical formulations, finite elements, boundary elements or a mix of these techniques. In the approach
proposed by Chopra and collaborators [2–4, 7], the reservoir is modeled analytically as a continuum fluid
region extending towards infinity in the upstream direction. When finite or boundary elements are used,
the reservoir has to be truncated at a finite distance and appropriate transmitting boundary conditions
have to be applied at the cutting boundaries to prevent reflection of spurious waves as discussed by the
authors in a previous work [13]. Some procedures were implemented in numerical codes specialized in
two- and three-dimensional analyses of concrete dams [9, 14], and some were validated against experi-
mental findings from in-situ forced-vibration tests [15–18]. Although such sophisticated techniques were
proven to efficiently handle many aspects of dam-reservoir interactions, their use requires appropriate
expertise and specialized software. For practical engineering applications, simplified procedures are still
needed to globally evaluate the seismic response of gravity dams, namely for preliminary design or safety
evaluation purposes [19–21].

The fundamental vibration period of dam-reservoir systems is a key factor in the assessment of their
dynamic or seismic behavior. Most seismic provisions and simplified procedures use the fundamen-
tal vibration period as an input parameter to determine seismic design accelerations and forces from
a site-specific earthquake response spectrum. It is therefore crucial to dispose of accurate and yet prac-
tical expressions to evaluate the fundamental period of gravity dams dynamically interacting with their
impounded reservoirs. Hatanaka [22] developed simplified expressions to estimate the fundamental vi-
bration period of dams with empty reservoirs. He approximated the dam geometry as a symmetrical
triangle and distinguished the cases where bending or shear effects are predominant in the dynamic re-
sponse of the dam. Considering analogy with beam theory, Okamoto [23] proposed simplified formulas
to estimate the fundamental vibration periods of dams with empty and full reservoirs. Chopra [2, 4] an-
alyzed several idealized triangular dam cross-sections to obtain an approximate fundamental vibration
period and corresponding mode shape of typical gravity dams with an empty reservoir. These standard
dynamic properties and related quantities were implemented in simplified earthquake response analyses
of gravity dams [19, 20]. To determine the fundamental vibration period of a dam including impounded
water effects, Chopra and collaborators [2–4, 7, 15] first obtained the frequency response curves char-

5
acterizing dam-reservoir vibrations, and then identified the fundamental vibration frequency as the one
corresponding to the first resonance on the curves. The authors found that hydrodynamic effects lengthen
the fundamental vibration period of gravity dams and the results obtained for standard dam cross-sections
were presented in figures and tables [19].

As mentioned above, although significant work has been devoted to investigate the effects of dam-water
interaction on the dynamic response of gravity dams, there is no available practical closed-form technique
to accurately estimate the fundamental vibration period of a gravity dam including hydrodynamic effects.
In this work, we propose simplified analytical expressions and a systematic procedure to rigourously
determine the fundamental period of vibrating dam-reservoir systems and corresponding added damping,
force and mass. The method includes the effects of dam geometry and flexibility, water compressibility
and varying reservoir level. Formulations assuming either incompressible or compressible impounded
water are developed. To assess the efficiency and accuracy of the proposed procedure, we validate it
against classical Westergaard added mass formulation as well as other advanced analytical and finite
element techniques. We finally illustrate how the proposed technique can be efficiently implemented in a
simplified and practical earthquake analysis of dam-reservoir systems.

2 Analytical formulation for vibrating dam-reservoir systems

2.1 Basic assumptions

The formulation described in this section was originally developed by Fenves and Chopra [7] to in-
vestigate earthquake excited gravity dams impounding semi-infinite rectangular-shape reservoirs. The
approach is based on a sub-structuring technique, where the dam is modeled using finite elements and
reservoir effects are accounted for analytically through hydrodynamic loads applied at dam upstream face.
The hydrodynamic pressures are obtained by first determining mode shapes of the dam with an empty
reservoir and then applying these mode shapes as boundary conditions to the solution of Helmholtz
equation that governs reservoir motion in the frequency domain. Bouaanani and Lu [24] showed that
this procedure to include dam-reservoir interaction yields excellent results when compared to techniques
where the reservoir is modeled numerically using potential-based fluid finite elements. The basic equa-
tions of the formulation are reviewed in this section considering compressible and incompressible water
assumptions.

To illustrate the dynamics of dam-reservoir systems, we consider a 2D gravity dam cross-section shown
in Fig. 1. The dam has a total height Hs and it impounds a semi-infinite reservoir of constant depth Hr .
A Cartesian coordinate system with axes x and y with origin at the heel of the structure is adopted and
the following main assumptions are made : (i) the dam and water are assumed to have a linear elastic
behavior; (ii) the dam foundation is assumed rigid; (iii) the water in the reservoir is assumed inviscid,
with its motion irrotational and limited to small amplitudes; and (iv) gravity surface waves are neglected.

6
 Figure 1. Dam-reservoir system.

2.2 Coupling hydrodynamic pressure and dam structural response

Considering a unit horizontal and harmonic exciting free-field ground motion ẍg (t) = eiωt , the hydrody-
namic pressure in the reservoir can be expressed in the frequency domain as p(x, y, t) = p̄(x, y, ω) eiωt ,
where ω denotes the exciting frequency, and p̄(x, y, ω) a complex-valued frequency response function
(FRF) obeying the classical Helmholtz equation

∂ 2 p̄ ∂ 2 p̄ ω 2
+ + p̄ = 0 (1)
∂x2 ∂y 2 Cr2

where Cr is the velocity of pressure waves in water. Fenves and Chopra [7] showed that hydrodynamic
pressure FRF p̄ can be decomposed as
Ns
X
p̄(x, y, ω) = p̄0 (x, y, ω) − ω 2 Z̄j (ω) p̄j (x, y, ω) (2)
j=1

in which p̄0 is the FRF for hydrodynamic pressure at rigid dam upstream face due to ground accelera-
(x)
tion, p̄j the FRF for hydrodynamic pressure due to horizontal acceleration ψj (0, y) of the dam upstream
(x)
face where ψj is the x–components of the j th structural mode shape ψ j , Z̄j the corresponding general-
ized coordinate and Ns the total number of mode shapes included in the analysis.

The complex FRFs p̄0 and p̄j can be expressed as the summation of Nr FRFs p̄0n and p̄jn corresponding

7
each to a reservoir mode n
Nr
X Nr
X
p̄0 (x, y, ω) = p̄0n (x, y, ω) ; p̄j (x, y, ω) = p̄jn (x, y, ω) (3)
n=1 n=1

FRFs p̄0n and p̄jn are given by

4ρr (−1)n eκn (ω)x

p̄0n (x, y, ω) = cos (λn y) (4)
π (2n − 1) κn (ω)
eκn (ω)x
p̄jn (x, y, ω) = −2ρr Ijn cos (λn y) (5)
κn (ω)
where ρr denotes water mass density and where the frequency-independent eigenvalues λn and terms κn
and Ijn are given by

(2n − 1) π
λn = (6)
2Hr
v
u
u
tλ2 (ω) − ω2
κn (ω) = n (7)
Cr2
1 Z Hr (x)
Ijn = ψj (0, y) cos (λn y) dy (8)
Hr 0

When water compressibility is neglected, i.e. Cr → +∞, Eq. (7) yields the frequency-independent
term κn = λn . Eqs. (4) and (5) simplify then to

8ρr Hr (−1)n λn x
p̄0n (x, y) = e cos (λn y) (9)
π 2 (2n − 1)2
4ρr Hr Ijn
p̄jn (x, y) = − eλn x cos (λn y) (10)
π (2n − 1)

Using modal superposition and mode shapes orthogonality, we show that the vector Z̄ of frequency-
dependent generalized coordinates Z̄j , j = 1 . . . Ns , can be obtained by solving the system of equations

S̄ Z̄ = Q̄ (11)

in which, for n = 1 . . . Ns and j = 1 . . . Ns

  Z Hr
S̄nj (ω) = ωn2 − ω 2 + 2 i ω ωn ξn Mn δnj + ω 2 p̄j (0, y, ω) ψn(x)(0, y) dy (12)
0
Z Hr
Q̄n (ω) = −Ln + p̄0 (0, y, ω) ψn(x)(0, y) dy (13)
0

with
Mn = ψ Tn M ψ n ; Ln = ψ Tn M 1 (14)

8
and where δnj is the Kronecker symbol, 1 is a column vector with ones when a horizontal translational
degree of freedom corresponds to the direction of earthquake excitation, and zero otherwise, M is the
dam mass matrix, ωn is the vibration frequency along mode shape ψ n , and ξn , Mn and Ln are the cor-
responding modal damping ratio, generalized mass and force, respectively. When mode shapes are also
mass-normalized, the generalized masses have unit values Mn = 1 for n = 1 . . . Ns . Eq. (2) can then be
applied to find FRFs for hydrodynamic pressure, and those for dam displacements and accelerations can
be expressed as
Ns
X Ns
X
(x) ¯ y, ω) = −ω 2 (x)
ū(x, y, ω) = ψj (x, y) Z̄j (ω) ; ü(x, ψj (x, y) Z̄j (ω) (15)
j=1 j=1

Ns
X Ns
X
(y) ¯ y, ω) = −ω 2 (y)
v̄(x, y, ω) = ψj (x, y) Z̄j (ω) ; v̈(x, ψj (x, y) Z̄j (ω) (16)
j=1 j=1

where ū and v̄ denote the horizontal and vertical displacements, respectively, ü¯ and v̈¯ the horizontal and
(x) (y)
vertical accelerations, respectively, ψj and ψj the x– and y–components of structural mode shape ψ j ,
and Ns the number of structural mode shapes included in the analysis.

3 Simplified formulation

3.1 Fundamental mode response analysis

As described in the previous section, a rigorous analysis of a dam-reservoir system requires the deter-
mination of several structural mode shapes of the dam with an empty reservoir. To investigate most
significant factors influencing dam seismic behavior, simplified procedures using only fundamental vi-
bration mode response have been developed and proven efficient for preliminary dam design and safety
evaluation [20]. Considering only the fundamental mode response, Eqs. (11) to (13) simplify to

−L1 − B0 (ω)
Z̄1 (ω) =  h i  h i (17)
−ω 2 M1 + Re B1 (ω) + iω C1 − ω Im B1 (ω) + K1

where the generalized earthquake force coefficient L1 , generalized mass M1 , generalized damping C1 ,
and generalized stiffness K1 of the Equivalent Single Degree of Freedom (ESDOF) system of the dam
with an empty reservoir are given by

L1 = ψ T1 M 1 ; M1 = ψ T1 M ψ 1 ; C1 = 2ξ1ω1 M1 ; K1 = ω12 M1 (18)

in which ξ1 is the fraction of critical damping at the fundamental vibration mode ψ 1 of the dam with an
empty reservoir, and ω1 its fundamental vibration frequency. A finite element analysis can be conducted
to obtain the generalized force L1 and generalized mass M1 from their discretized forms according to

9
Eq. (18). The following analytical expressions can also be used
ZZ
(x)
L1 = ρs (x, y) ψ1 (x, y) dxdy (19)
ZZ h i2 ZZ h i2
(x) (y)
M1 = ρs (x, y) ψ1 (x, y) dx dy + ρs (x, y) ψ1 (x, y) dx dy (20)

in which ρs is the mass density of the dam concrete. These equations can be simplified by approximating
the integration over the area of the dam by integration over its height [20] as
Z Hs Z Hs h i2
(x) (x)
L1 = µs (y) ψ1 (0, y) dy ; M1 = µs (y) ψ1 (0, y) dy (21)
0 0

where µs is the mass of the dam per unit height.

The complex-valued hydrodynamic terms B0 and B1 in Eq. (17) can be expressed as

Z Hr Nr
X
(x)
B0 (ω) = − p̄0 (0, y, ω) ψ1 (0, y) dy = B0n (ω) (22)
0 n=1

Z Hr Nr
X
(x)
B1 (ω) = − p̄1 (0, y, ω) ψ1 (0, y) dy = B1n (ω) (23)
0 n=1

in which
Z Hr
(x)
B0n (ω) = − p̄0n (0, y, ω) ψ1 (0, y) dy (24)
0
Z Hr
(x)
B1n (ω) = − p̄1n (0, y, ω) ψ1 (0, y) dy (25)
0

These parameters account for the effects of dam-reservoir interaction. As can be seen from Eq. (17), the
term B0 can be interpreted as an added force, the real part of B1 as an added mass and the imaginary part
of B1 as an added damping. Accordingly, Fenves and Chopra [7] showed that the seismic response of
a dam-reservoir system can be approximated by evaluating the generalized coordinate Z̄1 at the natural
vibration frequency ωr of the dam-reservoir system. At this frequency, hydrodynamic pressures p̄0 , p̄1 and
consequently hydrodynamic terms B0 and B1 are real, yielding from Eq. (17)

−L e
1
Z̄1 (ωr ) = (26)
−ωr M1 + i ωr Ce1 + ω12 M1
2 f

e , generalized mass M
where the generalized force L f and generalized damping C
e of the dam-reservoir
1 1 1
ESDOF system are obtained by modifying the parameters of the ESDOF system of the dam with an
empty reservoir as follows

e = L + B (ω )
L (27)
1 1 0 r
h i
f = M + Re B (ω ) = M + B (ω )
M (28)
1 1 1 r 1 1 r
h i
Ce1 = C1 − ωr Im B1 (ωr ) = C1 (29)

10
From Eq. (29), we may deduce the equivalent damping ratio ξer of the dam-reservoir ESDOF system as

Ce1
ξer = f
(30)
2ωr M 1

To develop analytical expressions for determining the fundamental vibration period of the dam including
the effects of impounded water, we assume that the x–component of the dam fundamental mode shape ψ 1
can be approximated as a cubic polynomial function
!2 !3
(x) y y y
ψ1 (0, y) = a1 + a2 + a3 (31)
Hs Hs Hs

where y is a coordinate varying along the height of the structure measured from its base. The coeffi-
cients a1 , a2 and a3 can be determined based on a finite element analysis of the dam monolith as illus-
trated in Fig. 2, or using the fundamental mode shape of a standard gravity dam section proposed by
Fenves and Chopra [19] as will be shown later.

Figure 2. Approximation of the fundamental mode shape of a gravity dam.

11
3.2 Simplified formulation of dam-reservoir interaction assuming incompressible water

Introducing Eqs. (6), (9), (10) and (31) into Eqs. (22) and (23), we show that hydrodynamic terms B0n
and B1n are real-valued and frequency-independent. They can be expressed as
h i
(−1)n 2×(−1)n Fn (η) − (2n − 1) π Gn (η)
Bb0n = 8ρr η 2 Hs2 (32)
(2n − 1)3 π 3
h i2
2×(−1)n Fn (η) − (2n − 1) π Gn (η)
Bb1n = 4ρr η 2 Hs2 (33)
(2n − 1)3 π 3

where the hat sign indicates quantities corresponding to the incompressible water case, η = Hr/Hs denotes
the ratio of reservoir level to dam height, and where functions Fn and Gn are given by
" # " #
8 2 24
Fn (η) = ηa1 + 1 − 2 2 η a2 + 1 − η 3 a3
(2n − 1) π (2n − 1)2 π 2
" # (34)
2
4η 24η
Gn (η) = − 2 2 a1 − a3
(2n − 1) π (2n − 1)2 π 2
Eq. (17) simplifies then to
b
−L1 − B 0
Z̄1 (ω) =   (35)
b + iωC + K
−ω 2 M1 + B1 1 1

It can be shown numerically that the generalized damping C1 has little effect on the fundamental vibra-
tion frequency ωr of the dam-reservoir system. Consequently, ωr can be approximated as the excitation
frequency corresponding to the resonance of the generalized coordinate Z̄1 in Eq. (35) with C1 = 0,
yielding  
ωr2 M1 + Bb1 − K1 = 0 (36)
where
Nr
X
Bb1 = Bb1n = 4ρr Hs2 Φ(η, Nr ) (37)
n=1
in which the function Φ(η, Nr ) is defined by
h i2
Nr
X 2×(−1)n Fn (η) − (2n − 1) π Gn (η)
Φ(η, Nr ) = η 2 (38)
n=1 (2n − 1)3 π 3

A sufficient number Nr of reservoir modes should be included to determine the sum Φ in Eq. (38). Figure 3
illustrates the variation of Φ as a function of reservoir height ratio η and number of included reservoir
modes Nr . We show numerically that the sum Φ converges towards a function ϕb depending only on
reservoir height ratio η
   
4
lim Φ(η, Nr ) = η γb1 a21 + γb2 a1 a2 η + γb3 a22 2
+ γb4 a1 a3 η + γb5 a2 a3 η + 3
γb6 a23 η 4
Nr →+∞
(39)
= ϕ(η)
b

12
where coefficients γb1 to γb6 are given in Table 1.

Figure 3. Variation of Φ and ϕb as a function of reservoir height ratio η and number of included reservoir
modes Nr : (a) η = 0.50 and (b) η = 1.00.

The limit ϕb is also shown in Fig. 3. Replacing into Eq. (36) yields the fundamental resonant frequency
and period of a dam-reservoir system with water compressibility neglected
s
ω1 4ρr Hs2 ϕ(η)
b
ωr = s ; Tr = T1 1+ (40)
4ρr Hs2 ϕ(η)
b M1
1+
M1
where T1 denotes the fundamental vibration period of the dam with an empty reservoir.

To obtain a simplified expression of the generalized coordinate Z̄1 of the dam-reservoir system at reso-
nant frequency ωr , a simplified expression of the hydrodynamic term Bb0 has to be found. When water

13
Table 1. Coefficients γbi and γi , i = 1, . . . , 6.
Incompressible water Compressible water

γb1 = 25.769 × 10−3 γ1 = 8.735 × 10−3

γb2 = 31.820 × 10−3 γ2 = 14.059 × 10−3
γb3 = 10.405 × 10−3 γ3 = 5.776 × 10−3
γb4 = 22.082 × 10−3 γ4 = 11.172 × 10−3
γb5 = 15.031 × 10−3 γ5 = 9.343 × 10−3
γb6 = 5.587 × 10−3 γ6 = 3.840 × 10−3

compressibility is neglected, we have according to Eq. (32)

Nr
X
Bb0 = b = 8ρ H 2 Θ(η, N )
B0n r s r (41)
n=1

where the function Θ(η, Nr ) is given by

h i
Nr
X (−1)n−1 2×(−1)n−1 Fn (η) + (2n − 1) π Gn (η)
Θ(η, Nr ) = η 2 (42)
n=1 (2n − 1)3 π 3

As for the function Φ, we show numerically that the sum Θ converges towards a function θb depending
only on reservoir height ratio η
 
b
lim Θ(η, Nr ) = η 3 ζb1 a1 + ζb2 a2 η + ζb3 a3 η 2 = θ(η) (43)
Nr →+∞

where the coefficients ζb1 to ζb3 are given in Table 2. The hydrodynamic term B
b can then be approximated
0
as
B b
b = 8ρ H 2 θ(η) (44)
0 r s

Table 2. Coefficients ζbi and ζi , i = 1, 2, 3.

Incompressible water Compressible water

ζb1 = 27.234 × 10−3 ζ1 = 3.795 × 10−3

ζb2 = 15.323 × 10−3 ζ2 = 3.105 × 10−3
ζb3 = 10.006 × 10−3 ζ3 = 2.500 × 10−3

Neglecting the influence of damping on the fundamental vibration frequency ωr of the dam-reservoir
system and using the analytical expressions developed above, the properties given in Eqs. (27), (28)

14
and (30) to characterize the dam-reservoir ESDOF system can now be obtained as

L b
e = L + 8ρ H 2 θ(η) (45)
1 1 r s

f = M + 4ρ H 2 ϕ(η) ω12
M1 1 r s b = M1 (46)
ωr2
C1 ωr
ξe1 = f
= ξ1 (47)
2ωr M1 ω1

3.3 Simplified formulation of dam-reservoir interaction considering water compressibil-

ity

Introducing Eqs. (4) to (6) and Eq. (31) into Eqs. (22) and (23), we show that the hydrodynamic terms B0n
and B1n are now complex-valued and frequency-dependent, and that they can be expressed as
h i
(−1)n 2×(−1)n Fn (η) − (2n − 1) π Gn (η)
B0n (ω) = 4ρr ηHs s (48)
2 (2n − 1)2 π 2 ω2
(2n − 1) π 2 −
4η 2 Hs2 Cr2
h i2
2×(−1)n Fn (η) − (2n − 1) π Gn (η)
B1n (ω) = 2ρr ηHs s (49)
2 (2n − 1)2 π 2 ω2
(2n − 1) π2 −
4η 2 Hs2 Cr2

As mentioned previously, the fundamental vibration frequency ωr of the dam-reservoir system can be ap-
proximated as the frequency corresponding to the resonance of the generalized coordinate Z̄1 in Eq. (17)
with C1 = 0, yielding in this case h i
ωr2 M1 + B1 (ωr ) − K1 = 0 (50)
Eq. (50) is more difficult to solve than Eq. (36) obtained assuming incompressible water, since the term B1
is now frequency-dependent. To circumvent this difficulty, we show that we can approximate the value
of hydrodynamic term B1 at the resonant frequency ωr as
Nr
X
B1 (ωr ) = B1,1 (ωr ) + B1n (0) (51)
n=2

where B1,1 (ωr ) is given by

h i2
2F1 (η) + π G1 (η)
B1,1 (ωr ) = 4ρr η 2 Hs2 s (52)
ω2
π3 1 − r2
ω0

15
in which ω0 = πCr/(2Hr ) denotes the fundamental vibration frequency of the full reservoir, and where F1
and G1 can be obtained from Eq. (34) with n = 1
! !
8 24
F1 (η) = ηa1 + 1 − 2 η 2 a2 + 1 − 2 η 3 a3
π π
! (53)
4η 24η 2
G1 (η) = − 2 a1 − 2 a3
π π

The value of B1n at ω = 0 is given by Eq. (49)

h i2
2×(−1)n Fn (η) − (2n − 1) π Gn (η)
B1n (0) = 4ρr η 2 Hs2 (54)
(2n − 1)3 π 3

Eq. (51) can then be rewritten as

(  2 )
η2
B1 (ωr ) = B1,1 (ωr ) + 4ρr Hs2 Φ(η, Nr ) − 3 2F1 (η) + π G1 (η) (55)
π

where Φ(η, Nr ) is given by Eq. (38). Considering the limit as Nr → +∞, we find that

B1 (ωr ) = B1,1 (ωr ) + 4ρr Hs2 ϕ(η) (56)

in which
 2
η2
ϕ(η) = lim Φ(η, Nr ) − 2F1 (η) + π G1 (η)
Nr →+∞ π3
 2
η2 (57)
= ϕ(η)
b − 3 2F1 (η) + π G1 (η)
π
   
4
=η γ1 a21 + γ2 a1 a2 η + γ3 a22 2
+ γ4 a1 a3 η + γ5 a2 a3 η +3
γ6 a23 η 4

We note that ϕ(η) has the same expression as ϕ(η) b in Eq. (39), but with coefficients γ1 to γ6 cor-
responding to the compressible water case as indicated in Table 1. To validate Eq. (56), Fig. 4 com-
pares the term 4ρr Hs2 ϕ(η) to the real and imaginary parts of the hydrodynamic term (B1 − B1,1 ) deter-
mined at frequency ratios ω/ω0 varying from 0 to 4. As can be seen, the approximation in Eq. (56) is
valid for frequency ratios ω/ω0 up to 1, and a fortiori for the dam-reservoir fundamental frequency ωr ,
since ωr /ω0 < 1. Substituting Eq. (56) into Eq. (50) and introducing the frequency ratios Rr = ωr /ω0
and R1 = ω1 /ω0 , we show that Eq. (50) can be rewritten under the form of a cubic equation to be solved
for χ = Rr2
A1 χ3 + A2 χ2 + A3 χ + A4 = 0 (58)

16
where
4ρr Hs2 ϕ(η)
A0 = 1 + (59)
M1
A1 = A20 (60)
( 2 )2
  2
4ρr η 2 Hs
A2 = −A0 A0 + 2R12 + 2F1 (η) + π G1 (η) (61)
M1 π 3
 
A3 = R12 2A0 + R12 (62)

A4 = −R14 (63)

Figure 4. Variation of the terms 4ρr Hs2 ϕ(η) and (B1 − B1,1 ) as a function of frequency ratio ω/ω0 and
reservoir height ratio η: (a) η = 0.50 and (b) η = 1.00.

The fundamental vibration frequency ωr = ω0 Rr and period Tr = 2π/ωr of the dam-reservoir system can
then be obtained by solving Eq. (58) numerically or analytically using Cardano’s formula. In the latter
case, Eq. (58) can be first reduced to
Γ3 + D1 Γ + D2 = 0 (64)

17
where
 2  3
1 A2 A3 1 A2 2 A2 A2 A3 A4
Γ=χ+ ; D1 = − ; D2 = − + (65)
3 A1 A1 3 A1 27 A1 3A21 A1
Eq. (64) has three solutions Γ1 , Γ2 and Γ3 that can be expressed as [25]

Γ1 = U + V ; Γ2 = τ U + τ 2 V ; Γ3 = τ 2 U + τ V (66)

where   √
D2 √ 1/3 1 D1 1 3
U= − + ∆ ; V =− ; τ =− +i (67)
2 3 U 2 2
and where ∆ denotes the discriminant
 3  2
D1 D2
∆= + (68)
3 2
We denote as Γ∗ the only real solution among Γ1 , Γ2 and Γ3 that satisfies
A2 A2
6 Γ∗ 6 R12 + (69)
3A1 3A1
The frequency ratio Rr and fundamental vibration period Tr of the dam-reservoir system are then given
by s
ωr A2 2π
Rr = = Γ∗ − ; Tr = s (70)
ω0 3A1 A 2
ω 0 Γ∗ −
3A1

Once the vibration frequency ωr is known, we can determine the properties of the dam-reservoir ESDOF
system as described in the previous section for the case of incompressible water. When water compress-
ibility is included, we show that the hydrodynamic term B0 (ωr) can be expressed as
Nr
X
B0 (ωr ) = B0,1 (ωr ) + B0n (0) (71)
n=2

where B0,1 (ωr ) is given by

h i
2F1 (η) + π G1 (η)
B0,1 (ωr ) = 8ρr η 2 Hs2 q (72)
π3 1 − Rr2

and where the value of B0n at ω = 0 is obtained from Eq. (48)

h i
2×(−1)n−1 Fn (η) + (2n − 1) π Gn (η)
B0n (0) = 8ρr η 2 Hs2 (73)
(2n − 1)3 π 3
Eq. (71) can then be rewritten as
(  )
η2
B0 (ωr ) = B0,1 (ωr ) + 8ρr Hs2 Θ(η, Nr) − 3 2F1 (η) + π G1 (η) (74)
π

18
where Θ(η, Nr) is given by Eq. (42). Considering the limit as Nr → +∞, we find that

B0 (ωr ) = B0,1 (ωr ) + 8ρr Hs2 θ(η) (75)

in which
 
η2
θ(η) = lim Θ(η, Nr ) − 3 2F1 (η) + π G1 (η)
Nr →+∞ π
 
b η2 (76)
= θ(η) − 2F1 (η) + π G1 (η)
π3
 
= η 3 ζ1 a1 + ζ2 a2 η + ζ3 a3 η 2

where coefficients ζ1 to ζ3 are given in Table 2. Neglecting the influence of damping on the fundamental
vibration frequency of the dam-reservoir system and using the analytical expressions developed above,
Eqs. (27), (28) and (30) become when water compressibility is included
 h i
 2F1 (η) + π G1 (η) 
e = L + 8ρ H 2 θ(η) + η 2
L q (77)
1 1 r s
 
π3 1 − Rr2

 h i2 

 2F1 (η) + π G1 (η)  
f = M + 4ρ H 2 ϕ(η) + η 2 ω12
M1 1 r s q = M1 (78)

 
 ωr2
π 3 1 − R2 r
ωr
ξe1 = ξ1 (79)
ω1

3.4 Application to the simplified earthquake analysis of gravity dams

The maximum response of a dam-reservoir ESDOF system to a horizontal earthquake ground motion can
be approximated by its static response under the effect of equivalent lateral forces f1 applied at the dam
upstream face and expressed per unit dam height as [20, 26]
e
L  n o
1 (x)
f1 (y) = f Sa Tr , ξe1 µs (y) ψ1 (0, y) − p̄1 (0, y, ωr)
M1
" ! # (80)
e
L   y y y2 3
1
= f Sa Tr , ξe1 µs (y) a1 + a2 2 + a3 3 − p̄1 (0, y, ωr)
M1 Hs Hs Hs
 
where Sa Tr , ξe1 is the pseudo-acceleration ordinate of the earthquake design spectrum at vibration pe-
riod Tr and for damping ratio ξe1 of the dam-reservoir ESDOF system described previously, and where the

19
hydrodynamic pressure p̄1 (0, y, ωr) can be expressed using a cubic mode shape approximation as
Nr
" #
X 2 × (−1)n Fn (η) − (2n − 1) π Gn (η) (2n − 1) π
p̄1 (0, y, ωr) = 2ρr v
u cos y (81)
n=1 u (2n − 1)2 π 2 ω2 2Hr
(2n − 1) π t − r2
4η 2 Hs2 Cr

in which Fn and Gn are given by Eq. (34), and the ratio of generalized force L e to generalized mass M
f
1 1
is obtained from Eqs. (77) and (78). If water compressibility is neglected, Eq. (81) simplifies to
Nr
" #
X 2 × (−1)n Fn (η) − (2n − 1) π Gn (η) (2n − 1) π
p̄1 (0, y, ωr) = 4ρr ηHs 2 2 cos y (82)
n=1 (2n − 1) π 2Hr

with Le and M f to be determined using Eqs. (45) and (46). We note that the minus sign in Eq. (80)
1 1
corresponds to the orientation of the system of axes shown in Fig. 5. We also assume that the fundamental
(x)
mode shape component ψ1 is positive as indicated on the same Figure.

Fenves and Chopra [19, 20] discussed the effects of higher vibration modes on dam earthquake response.
Using a static correction technique, this effect can be accounted for approximately by evaluating the static
response of the dam-reservoir ESDOF subjected to the lateral forces fsc applied at the dam upstream face
and expressed per unit dam height as
 " #
 L1 (x)
fsc (y) = ẍ(max) µ (y) 1 −
 s
ψ (0, y)
g
M1 1
(83)
" Z #
µs (y) (x) Hr 
(x)
− pb̄0 (0, y) + ψ (0, y) pb̄0 (0, y) ψ1 (0, y) dy
M1 1 0 

where ẍ(max)
g denotes the maximum ground acceleration, and pb̄0 (0, y) the real-valued, frequency-independent
hydrodynamic pressure applied on a rigid dam subjected to a unit ground acceleration and impounding
an incompressible water reservoir given by
Nr
" #
8ρr ηHs X (−1)n (2n − 1) π
pb̄0 (0, y) = 2 2 cos y (84)
π n=1 (2n − 1) 2ηHs

Assuming a cubic mode approximation, we show that Eq.(83) can be rewritten as

 " #" #! 
 L1 b y2 y3 
(max) 2 θ(η) y
fsc (y) = ẍg µs (y) 1 − + 8ρr Hs a1 + a2 2 + a3 3 − pb̄0 (0, y) (85)
 M1 M1 Hs Hs Hs 

The total earthquake response of the dam can then be determined by applying the SRSS rule to combine
response quantities associated with the fundamental and higher vibration modes [19, 20].

20
4 Dam models, analyses and results

4.1 Analyses conducted

In this section, we assess the effectiveness of the equations developed above in determining the funda-
mental mode response of gravity dams. To illustrate the analysis types conducted, we consider a dam
section with dimensions inspired from the tallest non-overflow monolith of Pine Flat dam [15]. The dam
cross-section is shown in Fig. 5 (a).

Figure 5. (a) Dam-reservoir system geometry; (b) Analysis type I: Finite element model; (c) Analysis
type II: analytical solution; (d) Analysis type III: Westergaard added mass formulation.

The following six types of analysis are conducted to determine the fundamental vibration frequency of
the dam-reservoir system:

– Analysis type I: a finite element analysis where both the dam and the reservoir are modeled using finite
elements. The software ADINA [27] is used to discretize the dam monolith into 9-node plane stress
finite elements. The reservoir is truncated at a large distance of 20Hr from the dam upstream face
to eliminate reflection of waves at the far reservoir upstream end. The 9-node potential-based finite

21
 elements programmed in ADINA [27] are used to model the reservoir. Fluid-structure interaction is
accounted for through special interface elements also included in the software. A finite element model
of the dam-reservoir system is shown in Fig. 5 (d). The performance of the potential-based formula-
tion and the fluid-structure interface elements was assessed in a previous work [24]. The method can
accurately account for fluid-structure interaction in dam-reservoir systems with a general geometry,
including when the dam upstream face is not vertical, which is for example the case of the slightly
inclined upstream face of the Pine Flat dam section. The results of this analysis will serve as our refer-
ence solution in the rest of the paper.

– Analysis type II: the analytical solution originally developed by Fenves and Chopra [7] and reviewed
in section 2. The same 9-node plane stress finite element model built for Type I analysis is used as
illustrated in Fig. 5 (c). The structural frequency response of the dam including hydrodynamic effects
is then determined using Eqs. (2) to (16). The fundamental frequency is identified next as that corre-
sponding to the first resonant structural response.

– Analysis type III: a finite element analysis of the Pine Flat dam where the reservoir hydrodynamic
loading is modeled approximately using Westergaard added mass formulation, assuming a rigid dam
with a vertical upstream face, impounding incompressible water [1]. The effect of the reservoir is
equivalent in this case to inertia forces generated by a body of water of parabolic shape moving back
and forth with the vibrating dam. The finite element model of the dam and the body of water are shown
in Fig. 5 (d). The added masse mi to be attached to a node i belonging to dam-reservoir interface can
be written as q
7
mi = ρr Vi Hr (Hr − yi ) (86)
8
where yi denotes the height of node i above the dam base and Vi the volume of water tributary to
node i. As previously, the software ADINA [27] is used to discretize the dam monolith into 9-node
plane stress finite elements.

– Analysis type IV: the new procedure proposed in this paper is applied using approximate parameters
(x)
L1 , M1 , ω1 and ψ1 proposed by Fenves and Chopra [19, 20]. The authors analyzed several standard
dam cross-sections and obtained the following conservative approximations for preliminary design
purposes: L1 = 0.13 Ms and M1 = 0.043 Ms , where Ms is the total mass of the dam monolith. Fenves
and Chopra [19, 20] also proposed to estimate the fundamental vibration frequency ω1 and period T1
of the dam with an empty reservoir as
√
2π Es 0.38 Hs
ω1 = ; T1 = √ (87)
0.38 Hs Es
where the dam concrete modulus of elasticity Es is expressed in MPa and Hs in meters to yield ω1
in rad/s and T1 in seconds. To develop a simplified earthquake analysis procedure, Fenves and Chopra [19,
20] used the standard fundamental mode shape given in Table 3. Applying the procedure illustrated in

22
 Fig. 2, this standard mode shape can be approximated using three points at elevations y1 = Hs /3, y2 =
2Hs /3 and y3 = Hs , yielding the coefficients a1 = 0.3535, a2 = −0.5455 and a3 = 1.1920. Eq. (31)
becomes then !2 !3
(x) y y y
ψ1 (0, y) = 0.3535 − 0.5455 + 1.1920 (88)
Hs Hs Hs

The resulting cubic interpolation is shown in Table 3. When water compressibility is neglected, Eq. (39)
simplifies to
h i
ϕ(η)
b = η 4 7.938 η 4 − 9.774 η 3 + 12.400 η 2 − 6.136 η + 3.220 × 10−3 (89)

after replacing the coefficients a1 to a3 by their values. Introducing M1 = 0.043 Ms and substituting
Eqs. (87) and (89) into Eq. (40) yields the dam-reservoir fundamental vibration frequency ωr and pe-
riod Tr when water compressibility is neglected. For example, considering a full reservoir, i.e. η = 1,
we obtain s
ω1 711.6Hs2
ωr = s ; Tr = T1 1 + (90)
711.6Hs2 Ms
1+
Ms
When water compressibility is included, replacing the coefficients a1 to a3 by their values into Eqs. (53)
and (57) yields
h i
ϕ(η) = η 4 5.456 η 4 − 6.075 η 3 + 6.426 η 2 − 2.711 η + 1.091 × 10−3 (91)

F1 (η) = 0.3535 η − 0.1033 η 2 − 1.7065 η 3 (92)

G1 (η) = −0.1433 η + 1.1748 η 3 (93)

The frequency ratio R1 can be approximated as

√
ω1 4η Es
R1 = = (94)
ω0 0.38 Cr
Coefficients A0 to A4 can be obtained using M1 = 0.043 Ms and substituting Eqs. (91) to (94) into
Eqs. (59) to (63). Eq. (58) is then solved for χ = Rr2 to obtain the fundamental vibration frequency ωr =
ω0 Rr and period Tr = 2π/ωr of the dam-reservoir system.

– Analysis type V: the new procedure proposed in this paper is applied using the approximate param-
(x)
eters L1 , M1 and ψ1 proposed by Fenves and Chopra [19, 20], but with the natural frequency ω1
obtained from a finite element analysis. All the equations described in the previous analysis Type IV
apply except for the frequency ratio R1 which now results from finite element analysis.

(x)
– Analysis type VI: the new procedure proposed in this paper is applied using parameters L1 , M1 , ψ1
and ω1 obtained from a finite element analysis of the dam section with an empty reservoir. A funda-
mental mode shape normalized with respect to the mass of the dam can be used, yielding a generalized

23
 mass M1 = 1. Applying the procedure illustrated in Fig. 2, the fundamental mode shape evaluated at
dam upstream face is interpolated using three points at elevations y1 = Hs /3, y2 = 2Hs /3 and y3 = Hs
to find the coefficients a1 to a3 in Eq. (31). Table 3 contains the original mode shape resulting from
finite element analysis of Pine Flat dam section as well as the cubic interpolation used. When water
compressibility is neglected, the resulting coefficients are introduced into Eq. (39) to obtain ϕ(η)
b and
then the dam-reservoir vibration frequency ωr using the generalized mass M1 and the fundamental vi-
bration frequency ω1 obtained from finite element analysis of the dam with an empty reservoir. When
water compressibility is included, coefficients a1 to a3 are introduced into Eqs. (57) and (53) to ob-
tain the parameters ϕ(η), F1 (η) and G1 (η). Coefficients A0 to A4 are determined next and Eq. (58) is
then solved for χ = Rr2 to obtain the vibration frequency of the dam-reservoir system as described in
section 3.

4.2 Validation of the proposed simplified formulation

The six analysis types described in the previous section are carried out to assess the effectiveness of the
method proposed in this paper. The Pine Flat dam section described previously is studied first. A mass
density ρs = 2400 kg/m3 and a Poisson’s ratio ν = 0.2 are assumed as concrete material properties.
To examine the influence of dam stiffness, two moduli of elasticity Es = 25 GPa and Es = 35 GPa are
considered. A water mass density ρr = 1000 kg/m3 is adopted. Both compressible and incompressible
water assumptions are investigated, with a pressure wave velocity of Cr = 1440 m/s in the former case.
We compute the period ratios Tr /T1 where Tr is the fundamental vibration period of the dam-reservoir
system obtained using any of the six analysis types described previously, and T1 is the reference funda-
mental vibration period determined using a finite element analysis of the dam with an empty reservoir.
Figures 6 and 7 illustrate the period ratios Tr /T1 obtained considering incompressible and compressible
water assumptions, respectively. Results for reservoir height ratios from η = 0.5 to 1.0 and two moduli of
elasticity Es = 25 GPa and Es = 35 GPa are given. Figures 6 and 7 also show bar charts representing the
following error estimator
Tr − Tr(FE)
ε= (FE)
(95)
Tr
where Tr(FE) denotes the reference fundamental vibration period obtained using a finite element analysis
of the dam-reservoir system, i.e. analysis type I.

First, it is apparent from the curves that the fundamental period predicted using finite elements, i.e.
analysis type I, and the analytical formulation proposed by Fenves and Chopra [7], i.e. analysis type II,
are very close for all height ratios and regardless of whether water is considered compressible or not.
This observation confirms the effectiveness of the analytical formulation even for dams with a slightly
inclined upstream face.

24
Table 3. Pine Flat dam fundamental mode shapes used.
(x) (x)
Normalized mode shape ψ1 (0, y)/ψ1 (0, Hs )

Fenves and Chopra [19] Finite element analysis

Original Cubic Original Cubic

y/Hs mode shape interpolation mode shape interpolation

1.00 1.000 1.000 1.000 1.000

0.95 0.866 0.866 0.875 0.871
0.90 0.735 0.745 0.752 0.755
0.85 0.619 0.638 0.640 0.650
0.80 0.530 0.544 0.543 0.556
0.75 0.455 0.461 0.461 0.472
0.70 0.389 0.389 0.391 0.398
0.65 0.334 0.327 0.331 0.333
0.60 0.284 0.273 0.279 0.277
0.55 0.240 0.228 0.233 0.228
0.50 0.200 0.189 0.194 0.186
0.45 0.165 0.157 0.159 0.150
0.40 0.135 0.130 0.129 0.120
0.35 0.108 0.108 0.102 0.094
0.30 0.084 0.089 0.080 0.073
0.25 0.065 0.073 0.060 0.056
0.20 0.047 0.058 0.044 0.042
0.15 0.034 0.045 0.030 0.030
0.10 0.021 0.031 0.019 0.019
0.05 0.010 0.016 0.010 0.009
0.00 0.000 0.000 0.000 0.000

25
Figure 6. Variation of period ratio Tr /T1 as a function of reservoir height ratio η assuming incompressible
water.

When water compressibility is neglected, Eq. (40) shows that the elasticity modulus of the dam has no
effet on the ratio Tr /T1 , a result that we confirmed numerically and analytically, i.e. using analysis types
I and II. Therefore, period ratios Tr /T1 for incompressible water are illustrated independently of the dam
elasticity modulus. Fig. 6 shows that analysis type III using Westergaard added mass predicts the funda-
mental frequency of the dam-reservoir system with a an error of about 12 per cent for a full reservoir in
the case of Pine Flat dam. Figs. 6 and 7 also clearly indicate that our simplified procedure, i.e. analysis
type VI, yields excellent results regardless of dam stiffness and compressible or incompressible water
assumptions. The results of the new simplified procedure remain in very good agreement when approx-
imate parameters are used instead of those obtained from finite element analysis of the dam section, i.e.
analysis types IV and V.

To investigate the influence of gravity dam cross-section geometry and dam stiffness on the accuracy of
the simplified procedure proposed in this paper, we analyse three typical gravity dam cross-sections with
heights varying from 90 m to 35 m as illustrated in Fig. 8. The three dams are denoted D1 to D3 from the
highest to the lowest. Finite element models of the dam sections and corresponding dam-reservoir systems
are built using the software ADINA [27]. The new simplified method is then applied using approximate
parameters, i.e. analysis types IV to V, as well as parameters resulting from finite element analyses of
each of the dam sections with an empty reservoir, i.e. analysis type VI. The period ratios Tr /T1 obtained
are illustrated in Fig. 9 considering reservoir height ratios from η = 0.5 to 1.0 and two moduli of elas-
ticity Es = 25 GPa and Es = 35 GPa. The different analyses are summarized in Table 4 for clarity purposes.

26
Figure 7. Variation of period ratio Tr /T1 as a function of reservoir height ratio η considering water com-
pressibility: (a) Es = 25 GPa and (b) Es = 35 GPa.

27
 Figure 8. Geometry and finite element models of gravity dam cross-sections D1, D2 and D3.

Table 4. Summary of analysis types conducted.

Gravity dam
Es = 25 GPa Es = 35 GPa
Water assumption Analysis Pine Flat D1 D2 D3 Pine Flat D1 D2 D3

Incompressible Type I x x x x x x x x
Type II x - - - x - - -
Type III x - - - x - - -
Type IV x x x x x x x x
Type V x x x x x x x x
Type VI x x x x x x x x

Compressible Type I x x x x x x x x
Type II x - - - x - - -
Type III - - - - - - - -
Type IV x x x x x x x x
Type V x x x x x x x x
Type VI x x x x x x x x

28
We first observe that the results of analysis types I and VI are almost identical for all the studied dam
sections independently of water compressibility or incompressibility assumptions, dam geometry and
stiffness. Analysis types IV and V yield satisfactory results for the 90-m high dam section D1. They are
less accurate however when applied to smaller dam sections D2 and D3. Analysis type IV introduces
large discrepancies because it uses approximate fundamental generalized force, generalized mass, mode
shape and vibration period that were mainly calibrated using higher standard dam sections [19, 20]. We
note that the fundamental period predictions are improved when an input fundamental vibration period
obtained from a finite element analysis of the dam with empty reservoir is used instead of Eq. (87), i.e.
analysis type V.

Based on the previous findings, we recommend to use the proposed simplified method according to
scheme of analysis type VI. The other schemes would provide appropriate results for high gravity dams,
while an increasing error is introduced for smaller dams. To asses the accuracy of the proposed method
in determining the damping ratio ξe1 of the dam-reservoir system ESDOF, Fig. 10 illustrates the variation
of this parameter as a function of reservoir height ratio η > 0.5 considering water compressibility, two
moduli of elasticity Es = 25 GPa and Es = 35 GPa and the four gravity dam cross-sections described
previously. In this figure, the results determined by applying the proposed method following the scheme
of analysis type VI are compared to those obtained using the classical method developed by Fenves
and Chopra [7] and reviewed in section 2. The curves clearly show that both techniques yield identical
damping ratios for the four dam monoliths.

Finally, denoting Fst = ρr gHr2 /2 the total hydrostatic force exerted on dam upstream face, we determine
the normalized equivalent lateral forces Hs f1 (y)/Fst considering a unit ordinate of pseudo-acceleration
spectrum, water compressibility, a full reservoir, i.e. η = 1, two moduli of elasticity and the four dams
cross-sections as before. Again, the resulting force distributions obtained using the classical and proposed
methods are practically coincident for the four dam monoliths studied as illustrated in Fig. 11.

5 Concluding remarks

This paper proposed an original practical method to evaluate the seismic response of gravity dams. We
first developed a simplified but yet a rigorous and practical formulation to determine the fundamen-
tal period of vibrating dam-reservoir systems and corresponding added damping, force and mass. The
new formulation includes the effects of dam geometry and flexibility, water compressibility and varying
reservoir level. The mathematical derivations of the method were provided considering both incompress-
ible and compressible water assumptions. In the former case, we proposed a closed-form expression to
determine the fundamental vibration period of a dam-reservoir system. When water compressibility is
considered, we showed that the fundamental vibration period of a dam-reservoir system can be obtained
by simply solving a cubic equation. Simplified expressions to compute the equivalent lateral earthquake
forces and the static correction forces are proposed. These forces are to be applied at the dam upstream

29
face to determine response quantities of interest, such as the stresses throughout the dam cross-section.

To assess the efficiency and accuracy of the proposed technique, several analysis types were applied to
dam cross-sections with various geometries and rigidities impounding reservoirs with different levels.
The following conclusions could be drawn from the comparison of the period predictions obtained from
the different analyses: (i) the analytical formulation of hydrodynamic effects yields accurate predictions
when compared to numerical results obtained by modeling the reservoir using potential-based finite el-
ements, (ii) the proposed simplified procedure gives excellent results when the fundamental generalized
earthquake force coefficient, generalized mass, mode shape and vibration period are directly obtained
from a finite element analysis of the dam with an empty reservoir, and (iii) the fundamental period pre-
dictions of the simplified procedure remain satisfactory for large dams while larger discrepancies are
observed for smaller ones when approximate parameters are used instead of those obtained from finite
element analysis. We also showed that the new procedure yields an excellent estimation of the equivalent
damping ratio and equivalent earthquake lateral forces. The proposed technique presents a significant
advantage over conventional Westergaard added-mass formulation, namely because it can directly ac-
count for dam flexibility and water compressibility, while Westergaard’s solution assumes that the dam
is rigid and water is incompressible. The analytical expressions developed and the procedure steps were
presented in a manner such that calculations could be easily implemented in a spreadsheet or program
for practical dynamic analysis of gravity dams. We clearly showed that the proposed procedure can be
used effectively for simplified evaluation of the vibration period and seismic response of gravity dams
irrespective of their geometry and stiffness.

Acknowledgements

The authors would like to acknowledge the financial support of the Natural Sciences and Engineering
Research Council of Canada (NSERC) and the Quebec Fund for Research on Nature and Technology
(FQRNT).

30
Figure 9. Variation of period ratio Tr /T1 as a function of reservoir height ratio η considering water com-
pressibility: (a) to (c) Dam D1; (d) to (f) Dam D2; and (g) to (i) Dam D3.

31
Figure 10. Variation of the damping ratio ξe1 as a function of reservoir height ratio η consider-
ing water compressibility: (a) and (b) Pine Flat dam; (c) and (d) Dam D1; (e) and (f) Dam D2;
and (g) and (h) Dam D3.

32
Figure 11. Normalized equivalent lateral earthquake forces corresponding to dam fundamental mode
response considering water compressibility: (a) and (b) Pine Flat dam; (c) and (d) Dam D1;
(e) and (f) Dam D2; and (g) and (h) Dam D3.

33
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